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Magnetic-field tunable photonic stop band in metallodielectric photonic crystals
Synthetic Metals 139 (2003) 705–709
Magnetic-field tunable photonic stop band in metallodielectric photonic crystals M. Golosovsky∗ , Y. Neve-Oz, D. Davidov The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Abstract We fabricated an artificial crystal consisting of a stack of containers with magnetizable ferromagnetic spheres. In the absence of external magnetic field the particles are in disordered state while in the presence of the field the particles self-assemble in almost a perfect hexagonal order. The degree of order is controlled by magnetic field. We study magnitude and phase of the frequency-dependent mm-wave transmission through the stack as a function of magnetic field. In the ordered state there are well-defined photonic stopbands separated by the regions where transmission is close to unity (“transparency windows”) while in the disordered state these regions mostly disappear. By varying magnetic field we achieve effective tuning of the mm-wave transmission through the stack. © 2003 Elsevier B.V. All rights reserved. Keywords: Photonic crystal; Stopband; Self-assembly; Tunability; Magnetic field
Photonic crystals are two-dimensional (2D) or threedimensional (3D) ordered structures which exhibit stopband, or even complete bandgap [1]. Many fabrication routes of such crystals have been reported recently and the important challenge now is tunability. This is usually achieved by varying the lattice constant or symmetry of the perfectly ordered crystal using different agents such as temperature, elastic stress and magnetic field [2–10]. We report here a novel photonic crystal where tunability is achieved by magnetic field induced order–disorder transition. Our building block is a transparent hexagonally-shaped plexiglas plate filled with 2 mm diameter magnetizable steel spheres which can move freely in lateral directions (Fig. 1). The motion of the spheres is limited by the walls made of magnetizable rods. The friction force between the spheres and the substrate is small but not negligible. The number of particles in each container corresponds to the perfect hexagonal packing, namely Z = 1 + 3s(s + 1), where s = 1, 2, 3, . . . . Several containers are arranged in a stack mounted inside the Helmholtz coils (Fig. 2). Perpendicular magnetic field of ∼10 mT magnetizes the spheres and induces magnetic interaction between them. The out-of-plane attraction between the spheres is too weak to induce ordering across the layers, while the in-plane repulsion is strong enough to drive each 2D-array into a well-ordered ∗ Corresponding author. Tel.: +972-2-658-5139/6551; fax: +972-2-5617-805. E-mail address: [email protected] (M. Golosovsky).
0379-6779/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0379-6779(03)00331-X
“crystalline” state with almost constant density (Fig. 1a). Intermediate field (1-10 mT) drives each array into homogeneous “amorphous” state (Fig. 1b) exhibiting short-range rather than long-range order. In the absence of magnetic field this structure exhibits disordered “aggregated” state (Fig. 1c) with inhomogeneous in-plane particle density. The right panel of Fig. 1 shows Fourier transform of the corresponding images. Note sharp spots in the “crystalline” state (Fig. 1a); spots and a ring in the “amorphous” state (Fig. 1b); a wide “diffuse” ring in the “aggregated” state (Fig. 1c). Similar systems consisting of colloidal magnetic particles on liquid surface were studied recently [12,13]. The mm-wave transmission through our structures was measured using a HP850C Vector Network Analyzer and two standard gain horn antennae (Fig. 2) to which we attached home-made collimating Teflon lenses. We studied transmission in the 20–50 GHz frequency range as a function of number of layers, layer spacing, and magnetic field. The measurements were performed as follows. We measured frequency dependence of the mm-wave transmission at fixed value of magnetic field and then moved to the next value of the field. In between the measurements we either sent a short and strong magnetic pulse or vibrated the stack in order to erase the memory of the previous state. At small values of magnetic field, when the particles are only partially ordered, there are many particular realizations of the configuration corresponding to a certain value of the field. The mm-wave transmission through these configurations differ only in minor de-
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M. Golosovsky et al. / Synthetic Metals 139 (2003) 705–709
Fig. 1. Particle arrangement in container. The container consists of two thin plexiglas plates and an array of 331 steel spheres of 2 mm diameter. The side of the container is 3 cm. The in-plane lattice constant in the ordered state of the array is 2.9 mm. The left row shows real space pictures and the right row shows corresponding 2D-Fourier transform. Note gradual transition from disordered to ordered state upon increasing magnetic field.
tails (the magnitude and position of small sharp peaks in Figs. 3 and 4). Fig. 3 shows mm-wave transmission through our stack for different values of external magnetic field. At strong enough fields (above 10 mT) the particles self-assemble into the“crystalline” state (Fig. 1a) exhibiting a well-defined photonic stopband at 25–45 GHz (Fig. 3). At intermediate fields corresponding to the “amorphous” state (Fig. 1b), the stopband is still present, although its edges are smeared. For a very weak field (below 1 mT) corresponding to the “aggregated” state (Fig. 1c), the stopband is completely smeared and the transmission is low in the entire frequency range. Fig. 4 shows mm-wave transmission through a slightly different stack which has the same in-plane number of particles but increased layer spacing. Here, the stopband
in the “crystalline” state is located at lower frequencies (20–36 GHz), and the wide region at 35–48 GHz where transmission is close to unity (“transparency window”) is clearly observed. We can even notice the development of the second photonic stopband above 48 GHz. [The corresponding region for the stack shown in Fig. 3 is above the frequency range of our setup.] In the “amorphous” state the “transparency window” becomes narrower and the transmission there decreases. In the “aggregated” state the transmission in the “transparency window” further decreases but is still higher than at neighboring regions—this is a so-called Bragg remnant which is well-known in the context of wave transmission through disordered systems [11]. The effect of the order–disorder transition is most pronounced at the stopband edges while the transmission in
M. Golosovsky et al. / Synthetic Metals 139 (2003) 705–709
707
Fig. 2. Measurement setup. Standard gain microwave horns are connected to HP 8510C Vector Network Analyzer and are terminated by two home-made collimating Teflon lenses. Magnetic field is perpendicular to the layers.
the center of the stopband is hardly affected. Fig. 5 shows magnetic field dependence of the mm-wave transmission at the lower edge of the second photonic bandgap. The magnetic-field-dependent mm-wave transmission at fixed frequency is accounted for by the empirical relation T(H ) 4 = e−C1 /(H +C2 ) T(H → ∞)
(1)
where C1 and C2 are fitting parameters. Our results may be accounted by the Bragg reflector model [14]. It assumes a periodic two-component multilayer [AB]N with N unit cells. The sublayers A and B are uniform
Fig. 4. Transmission through the eight-layer stack of containers for different strengths of magnetic field. The interlayer distance is 4.75 mm. Note the first photonic stopband at 20–36 GHz and the second photonic stopband above 48 GHz.
and have thickness di , refraction index ni , and admittance Yi which are related to the dielectric constant i and magnetic permeability µi as follows: Yi = (i /µi )1/2 , ni = (i µi )1/2 . For normal incidence, the first gap of such multilayer occurs at c f0 = (2) 2(nA dA + nB dB ) where f0 is the midgap frequency and d = dA + dB is the unit cell thickness in the direction of wave propagation.
Fig. 3. Transmission through the eight-layer stack of containers for different strengths of magnetic field. The interlayer distance is 3.75 mm. The value of magnetic field is shown at each curve. Note photonic stopband at 25–45 GHz.
Fig. 5. Transmission through the array of Fig. 4 at f = 47.75 GHz as a function of magnetic field. Continuous curve shows theoretical estimate according to Eq. (1).
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M. Golosovsky et al. / Synthetic Metals 139 (2003) 705–709
We assume that the sublayer A is a layer of conducting spheres while the sublayer B is the spacing between the layers. We consider each layer as uniform medium and neglect the difference between the dielectric constants of the air ( = 1) and plexiglas ( = 1.6). We assume also that dA = 2r, where r is the radius of a sphere, and dB = d − dA . We assume also nB = 1, YB = Y0 . The refractive index and admittance of the sublayer A are found from the effective-medium approximation [15]. In this framework n and Y are determined only by the particle density. The [15] yields for the array of perfectly conducting spheres nA = (1 + 2p)1/2 (1 + p/2)−1/2 where p is the particle filling factor. The effective-medium approximation and the Bragg reflector model correctly account for position, width, and depth of the stopband in our stacks in the ordered state [16]. The effect of disorder can be explained using the same framework. Here, we exploit the similarity between our results and X-ray propagation in disordered crystals on one hand; and optical properties of liquids in the vicinity of the critical point, on another hand [17]. Let us analyze the frequency range corresponding to the “transparency window”. Here the sublayer of spheres represents a half-wavelength plate, in other words nA dA ≈ λ/2. Transmission through the stack of such plates is close to unity due to destructive interference between the waves reflected from each layer. We relate the decrease of transmission in the disordered state to the fluctuations of the particle density in each layer, p. The refraction index of each sublayer fluctuates accordingly, nA = (∂n/∂p)p. Therefore, each sublayer locally deviates from the condition of a half-wavelength plate, destructive interference upon reflection is incomplete and the transmission decreases correspondingly. We describe this decrease by the Debye–Waller factor [17] T(H) 2 = e−N(kd n) T(H → ∞)
Eq. (3), we come to Eq. (1). The term C2 in Eq. (1) is required to lift the divergence when H → 0, since the density fluctuations are limited by the finite size of the spheres. The Eq. (1) correctly describes magnetic field dependence of the transmission in the “transparency window” and at the stopband edges (Fig. 5). The weaker magnetic field dependence of the mm-wave transmission in the center of the stopband is accounted for by a more sophisticated model, which takes into account the change of reflectivity of each layer due to fluctuations. Transmission through our stacks in the disordered state is very small due to large density fluctuations, similar to optical transmission through the liquid at the critical point [17]. Magnetic field suppresses density fluctuations and thus makes the material more transparent. Similar mechanism has been recently realized in the context of liquid crystals, where fluctuations in the optical birefringerence are controlled by electric field [18]. By scaling down our approach to the optical range, we envisage fabrication of tunable 2D photonic crystals for the optical and infrared ranges, based on surface wave propagation. If such surface is covered with the liquid layer containing movable magnetic particles, the propagation of the surface waves can be effectively monitored by magnetic field. A similar idea, based on particle reorientation rather than motion in magnetic field, has been theoretically suggested by Figotin et al. [19]. In conclusion, we demonstrate a photonic bandgap material, where transmission through which can be controlled by magnetic field. This concept may be useful for 2D photonic crystals and surface wave propagation. This work was supported by the VW Foundation, Israeli Science Foundation, and Israeli Ministry of Science and Arts. We are grateful to V. Freilikher and L. Shvartsman for valuable discussions.
(3)
where N is the number of layers and k is the wavevector. Magnetic field suppresses density fluctuations. Indeed, in the completely ordered state the distances between the particles are equal, the particle density may be considered uniform across the layer, and the fluctuations are absent. If some particle moves away from its equilibrium position, magnetic repulsion from the neighbors pushes its back. This occurs only when the restoring magnetic force Fmagn overcomes the limit set by dry friction, namely Fmagn ∼ (∂2 Umagn /∂a2 )|a| ≥ Ffr , where Umagn is the pair interaction energy, a is the in-plane lattice constant, and Ffr is the maximum friction force between a sphere and the supporting plate. Magnetic pair interaction energy is Umagn ∼ M 2 /a3 , where M is the magnetic moment of the sphere given by M = χH where χ is magnetic susceptibility. Finally, we find that the magnetic field suppresses fluctuations exceeding |a|max = const./H 2 , while the fluctuations with |a| < |a|max are not suppressed. Since p/p = −2(a/a), then (nA )2 ∼ (amax )2 ∼ 1/H 4 . Using this relation and
References [1] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals Molding the Flow of Light, Princeton University Press, NJ, (1995). [2] K. Busch, S. John, Phys. Rev. E 58 (1998) 3896. [3] P. Halevi, F. Ramos-Mendieta, Phys. Rev. Lett. 85 (2000) 1875. [4] S. Kim, V. Gopalan, Phys. Rev. Lett. 63 (2001) 5. [5] B. Gates, Y.N. Xia, Advanced Materials 13 (2001) 1605. [6] B.A. Grzybowski, H.A. Stone, G.W. Whitesides, Nature 405 (2000) 1033. [7] W.J. Wen, L.Y. Zhang, P. Sheng, Phys. Rev. Lett. 85 (2000) 5464. [8] X.L. Xu, G. Friedman, K.D. Humfeld, S.A. Majetich, S.A. Asher, Adv. Mater. 13 (2001) 1681. [9] E.L. Bizdoaca, M. Spasova, M. Farle, M. Hilgendorf, F. Caruso, J. Magn. Magn. Mater. 240 (2002) 44. [10] Y. Saado, M. Golosovsky, D. Davidov, A. Frenkel, Phys. Rev. B 66 (2002) 195108. [11] C. Martin de Sterke, R.C. McPhedran, Phys. Rev. B 47 (1993) 7780. [12] K. Zahn, R. Lenke, G. Maret, Phys. Rev. Lett. 82 (1999) 2721. [13] R. Bubeck, C. Bechinger, S. Neser, P. Leiderer, Phys. Rev. Lett. 82 (1999) 3364.
M. Golosovsky et al. / Synthetic Metals 139 (2003) 705–709 [14] A. Yariv, P. Yeh, Optical Waves in Crystals, Wiley, 1984, NY (Chapter 6). [15] A.K. Sarychev, R.C. McPhedran, V.M. Shalaev, Phys. Rev. B 62 (2000) 8531. [16] Y. Neve-Oz, M. Golosovsky, D. Davidov, A. Frenkel, Phys. Rev. E, in press.
709
[17] L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii, Electrodynamics of Continuous Media, Pergamon Press, London, 1984 (Chapter 15). [18] O. Levy, Phys. Rev. E 61 (2000) 5385. [19] A. Figotin, Y.A. Godin, I. Vitebsky, Phys. Rev. B 57 (1998) 2841.
Magnetic-field tunable photonic stop band in metallodielectric photonic crystals M. Golosovsky∗ , Y. Neve-Oz, D. Davidov The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Abstract We fabricated an artificial crystal consisting of a stack of containers with magnetizable ferromagnetic spheres. In the absence of external magnetic field the particles are in disordered state while in the presence of the field the particles self-assemble in almost a perfect hexagonal order. The degree of order is controlled by magnetic field. We study magnitude and phase of the frequency-dependent mm-wave transmission through the stack as a function of magnetic field. In the ordered state there are well-defined photonic stopbands separated by the regions where transmission is close to unity (“transparency windows”) while in the disordered state these regions mostly disappear. By varying magnetic field we achieve effective tuning of the mm-wave transmission through the stack. © 2003 Elsevier B.V. All rights reserved. Keywords: Photonic crystal; Stopband; Self-assembly; Tunability; Magnetic field
Photonic crystals are two-dimensional (2D) or threedimensional (3D) ordered structures which exhibit stopband, or even complete bandgap [1]. Many fabrication routes of such crystals have been reported recently and the important challenge now is tunability. This is usually achieved by varying the lattice constant or symmetry of the perfectly ordered crystal using different agents such as temperature, elastic stress and magnetic field [2–10]. We report here a novel photonic crystal where tunability is achieved by magnetic field induced order–disorder transition. Our building block is a transparent hexagonally-shaped plexiglas plate filled with 2 mm diameter magnetizable steel spheres which can move freely in lateral directions (Fig. 1). The motion of the spheres is limited by the walls made of magnetizable rods. The friction force between the spheres and the substrate is small but not negligible. The number of particles in each container corresponds to the perfect hexagonal packing, namely Z = 1 + 3s(s + 1), where s = 1, 2, 3, . . . . Several containers are arranged in a stack mounted inside the Helmholtz coils (Fig. 2). Perpendicular magnetic field of ∼10 mT magnetizes the spheres and induces magnetic interaction between them. The out-of-plane attraction between the spheres is too weak to induce ordering across the layers, while the in-plane repulsion is strong enough to drive each 2D-array into a well-ordered ∗ Corresponding author. Tel.: +972-2-658-5139/6551; fax: +972-2-5617-805. E-mail address: [email protected] (M. Golosovsky).
0379-6779/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0379-6779(03)00331-X
“crystalline” state with almost constant density (Fig. 1a). Intermediate field (1-10 mT) drives each array into homogeneous “amorphous” state (Fig. 1b) exhibiting short-range rather than long-range order. In the absence of magnetic field this structure exhibits disordered “aggregated” state (Fig. 1c) with inhomogeneous in-plane particle density. The right panel of Fig. 1 shows Fourier transform of the corresponding images. Note sharp spots in the “crystalline” state (Fig. 1a); spots and a ring in the “amorphous” state (Fig. 1b); a wide “diffuse” ring in the “aggregated” state (Fig. 1c). Similar systems consisting of colloidal magnetic particles on liquid surface were studied recently [12,13]. The mm-wave transmission through our structures was measured using a HP850C Vector Network Analyzer and two standard gain horn antennae (Fig. 2) to which we attached home-made collimating Teflon lenses. We studied transmission in the 20–50 GHz frequency range as a function of number of layers, layer spacing, and magnetic field. The measurements were performed as follows. We measured frequency dependence of the mm-wave transmission at fixed value of magnetic field and then moved to the next value of the field. In between the measurements we either sent a short and strong magnetic pulse or vibrated the stack in order to erase the memory of the previous state. At small values of magnetic field, when the particles are only partially ordered, there are many particular realizations of the configuration corresponding to a certain value of the field. The mm-wave transmission through these configurations differ only in minor de-
706
M. Golosovsky et al. / Synthetic Metals 139 (2003) 705–709
Fig. 1. Particle arrangement in container. The container consists of two thin plexiglas plates and an array of 331 steel spheres of 2 mm diameter. The side of the container is 3 cm. The in-plane lattice constant in the ordered state of the array is 2.9 mm. The left row shows real space pictures and the right row shows corresponding 2D-Fourier transform. Note gradual transition from disordered to ordered state upon increasing magnetic field.
tails (the magnitude and position of small sharp peaks in Figs. 3 and 4). Fig. 3 shows mm-wave transmission through our stack for different values of external magnetic field. At strong enough fields (above 10 mT) the particles self-assemble into the“crystalline” state (Fig. 1a) exhibiting a well-defined photonic stopband at 25–45 GHz (Fig. 3). At intermediate fields corresponding to the “amorphous” state (Fig. 1b), the stopband is still present, although its edges are smeared. For a very weak field (below 1 mT) corresponding to the “aggregated” state (Fig. 1c), the stopband is completely smeared and the transmission is low in the entire frequency range. Fig. 4 shows mm-wave transmission through a slightly different stack which has the same in-plane number of particles but increased layer spacing. Here, the stopband
in the “crystalline” state is located at lower frequencies (20–36 GHz), and the wide region at 35–48 GHz where transmission is close to unity (“transparency window”) is clearly observed. We can even notice the development of the second photonic stopband above 48 GHz. [The corresponding region for the stack shown in Fig. 3 is above the frequency range of our setup.] In the “amorphous” state the “transparency window” becomes narrower and the transmission there decreases. In the “aggregated” state the transmission in the “transparency window” further decreases but is still higher than at neighboring regions—this is a so-called Bragg remnant which is well-known in the context of wave transmission through disordered systems [11]. The effect of the order–disorder transition is most pronounced at the stopband edges while the transmission in
M. Golosovsky et al. / Synthetic Metals 139 (2003) 705–709
707
Fig. 2. Measurement setup. Standard gain microwave horns are connected to HP 8510C Vector Network Analyzer and are terminated by two home-made collimating Teflon lenses. Magnetic field is perpendicular to the layers.
the center of the stopband is hardly affected. Fig. 5 shows magnetic field dependence of the mm-wave transmission at the lower edge of the second photonic bandgap. The magnetic-field-dependent mm-wave transmission at fixed frequency is accounted for by the empirical relation T(H ) 4 = e−C1 /(H +C2 ) T(H → ∞)
(1)
where C1 and C2 are fitting parameters. Our results may be accounted by the Bragg reflector model [14]. It assumes a periodic two-component multilayer [AB]N with N unit cells. The sublayers A and B are uniform
Fig. 4. Transmission through the eight-layer stack of containers for different strengths of magnetic field. The interlayer distance is 4.75 mm. Note the first photonic stopband at 20–36 GHz and the second photonic stopband above 48 GHz.
and have thickness di , refraction index ni , and admittance Yi which are related to the dielectric constant i and magnetic permeability µi as follows: Yi = (i /µi )1/2 , ni = (i µi )1/2 . For normal incidence, the first gap of such multilayer occurs at c f0 = (2) 2(nA dA + nB dB ) where f0 is the midgap frequency and d = dA + dB is the unit cell thickness in the direction of wave propagation.
Fig. 3. Transmission through the eight-layer stack of containers for different strengths of magnetic field. The interlayer distance is 3.75 mm. The value of magnetic field is shown at each curve. Note photonic stopband at 25–45 GHz.
Fig. 5. Transmission through the array of Fig. 4 at f = 47.75 GHz as a function of magnetic field. Continuous curve shows theoretical estimate according to Eq. (1).
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M. Golosovsky et al. / Synthetic Metals 139 (2003) 705–709
We assume that the sublayer A is a layer of conducting spheres while the sublayer B is the spacing between the layers. We consider each layer as uniform medium and neglect the difference between the dielectric constants of the air ( = 1) and plexiglas ( = 1.6). We assume also that dA = 2r, where r is the radius of a sphere, and dB = d − dA . We assume also nB = 1, YB = Y0 . The refractive index and admittance of the sublayer A are found from the effective-medium approximation [15]. In this framework n and Y are determined only by the particle density. The [15] yields for the array of perfectly conducting spheres nA = (1 + 2p)1/2 (1 + p/2)−1/2 where p is the particle filling factor. The effective-medium approximation and the Bragg reflector model correctly account for position, width, and depth of the stopband in our stacks in the ordered state [16]. The effect of disorder can be explained using the same framework. Here, we exploit the similarity between our results and X-ray propagation in disordered crystals on one hand; and optical properties of liquids in the vicinity of the critical point, on another hand [17]. Let us analyze the frequency range corresponding to the “transparency window”. Here the sublayer of spheres represents a half-wavelength plate, in other words nA dA ≈ λ/2. Transmission through the stack of such plates is close to unity due to destructive interference between the waves reflected from each layer. We relate the decrease of transmission in the disordered state to the fluctuations of the particle density in each layer, p. The refraction index of each sublayer fluctuates accordingly, nA = (∂n/∂p)p. Therefore, each sublayer locally deviates from the condition of a half-wavelength plate, destructive interference upon reflection is incomplete and the transmission decreases correspondingly. We describe this decrease by the Debye–Waller factor [17] T(H) 2 = e−N(kd n) T(H → ∞)
Eq. (3), we come to Eq. (1). The term C2 in Eq. (1) is required to lift the divergence when H → 0, since the density fluctuations are limited by the finite size of the spheres. The Eq. (1) correctly describes magnetic field dependence of the transmission in the “transparency window” and at the stopband edges (Fig. 5). The weaker magnetic field dependence of the mm-wave transmission in the center of the stopband is accounted for by a more sophisticated model, which takes into account the change of reflectivity of each layer due to fluctuations. Transmission through our stacks in the disordered state is very small due to large density fluctuations, similar to optical transmission through the liquid at the critical point [17]. Magnetic field suppresses density fluctuations and thus makes the material more transparent. Similar mechanism has been recently realized in the context of liquid crystals, where fluctuations in the optical birefringerence are controlled by electric field [18]. By scaling down our approach to the optical range, we envisage fabrication of tunable 2D photonic crystals for the optical and infrared ranges, based on surface wave propagation. If such surface is covered with the liquid layer containing movable magnetic particles, the propagation of the surface waves can be effectively monitored by magnetic field. A similar idea, based on particle reorientation rather than motion in magnetic field, has been theoretically suggested by Figotin et al. [19]. In conclusion, we demonstrate a photonic bandgap material, where transmission through which can be controlled by magnetic field. This concept may be useful for 2D photonic crystals and surface wave propagation. This work was supported by the VW Foundation, Israeli Science Foundation, and Israeli Ministry of Science and Arts. We are grateful to V. Freilikher and L. Shvartsman for valuable discussions.
(3)
where N is the number of layers and k is the wavevector. Magnetic field suppresses density fluctuations. Indeed, in the completely ordered state the distances between the particles are equal, the particle density may be considered uniform across the layer, and the fluctuations are absent. If some particle moves away from its equilibrium position, magnetic repulsion from the neighbors pushes its back. This occurs only when the restoring magnetic force Fmagn overcomes the limit set by dry friction, namely Fmagn ∼ (∂2 Umagn /∂a2 )|a| ≥ Ffr , where Umagn is the pair interaction energy, a is the in-plane lattice constant, and Ffr is the maximum friction force between a sphere and the supporting plate. Magnetic pair interaction energy is Umagn ∼ M 2 /a3 , where M is the magnetic moment of the sphere given by M = χH where χ is magnetic susceptibility. Finally, we find that the magnetic field suppresses fluctuations exceeding |a|max = const./H 2 , while the fluctuations with |a| < |a|max are not suppressed. Since p/p = −2(a/a), then (nA )2 ∼ (amax )2 ∼ 1/H 4 . Using this relation and
References [1] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals Molding the Flow of Light, Princeton University Press, NJ, (1995). [2] K. Busch, S. John, Phys. Rev. E 58 (1998) 3896. [3] P. Halevi, F. Ramos-Mendieta, Phys. Rev. Lett. 85 (2000) 1875. [4] S. Kim, V. Gopalan, Phys. Rev. Lett. 63 (2001) 5. [5] B. Gates, Y.N. Xia, Advanced Materials 13 (2001) 1605. [6] B.A. Grzybowski, H.A. Stone, G.W. Whitesides, Nature 405 (2000) 1033. [7] W.J. Wen, L.Y. Zhang, P. Sheng, Phys. Rev. Lett. 85 (2000) 5464. [8] X.L. Xu, G. Friedman, K.D. Humfeld, S.A. Majetich, S.A. Asher, Adv. Mater. 13 (2001) 1681. [9] E.L. Bizdoaca, M. Spasova, M. Farle, M. Hilgendorf, F. Caruso, J. Magn. Magn. Mater. 240 (2002) 44. [10] Y. Saado, M. Golosovsky, D. Davidov, A. Frenkel, Phys. Rev. B 66 (2002) 195108. [11] C. Martin de Sterke, R.C. McPhedran, Phys. Rev. B 47 (1993) 7780. [12] K. Zahn, R. Lenke, G. Maret, Phys. Rev. Lett. 82 (1999) 2721. [13] R. Bubeck, C. Bechinger, S. Neser, P. Leiderer, Phys. Rev. Lett. 82 (1999) 3364.
M. Golosovsky et al. / Synthetic Metals 139 (2003) 705–709 [14] A. Yariv, P. Yeh, Optical Waves in Crystals, Wiley, 1984, NY (Chapter 6). [15] A.K. Sarychev, R.C. McPhedran, V.M. Shalaev, Phys. Rev. B 62 (2000) 8531. [16] Y. Neve-Oz, M. Golosovsky, D. Davidov, A. Frenkel, Phys. Rev. E, in press.
709
[17] L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii, Electrodynamics of Continuous Media, Pergamon Press, London, 1984 (Chapter 15). [18] O. Levy, Phys. Rev. E 61 (2000) 5385. [19] A. Figotin, Y.A. Godin, I. Vitebsky, Phys. Rev. B 57 (1998) 2841.